Repeating decimal

recurring decimalrepeatrepetendrepeating fractionperiod lengthrecurringrepeating decimals0.33333...2.algorithm for positive bases
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.wikipedia
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Rational number

rationalrational numbersrationals
It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). In order to convert a rational number represented as a fraction into decimal form, one may use long division.
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over.

0.999...

0.999…Proof that 0.999... equals 10.
[[0.999...|]] and are two examples of this.
In mathematics, 0.999... (also written 0., among other ways) denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).

Decimal

base 10decimal systemdecimal fraction
Every terminating decimal representation can be written as a decimal fraction, a fraction whose divisor is a power of 10 (e.g. ); it may also be written as a ratio of the form k⁄2 n 5 m (e.g. ). However, every number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit 9.
A repeating decimal is an infinite decimal that after some place repeats indefinitely the same sequence of digits (for example

Pi

ππ\pi
Examples of such irrational numbers are the square root of 2 and [[pi|]].
Being an irrational number, cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern).

Irrational number

irrationalirrational numbersirrationality
Any number that cannot be expressed as a ratio of two integers is said to be irrational.
It can be shown that irrational numbers, when expressed in a positional numeral system (e.g. as decimal numbers, or with any other natural basis), do not terminate, nor do they repeat, i.e., do not contain a subsequence of digits, the repetition of which makes up the tail of the representation.

142,857

1428571⁄7 (number)2/7 (number)
See also the article 142,857 for more properties of this cyclic number.
142857, the six repeating digits of 1⁄7, 0., is the best-known cyclic number in base 10.

Vinculum (symbol)

vinculum
Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal is a significant exception and reflects the original usage.

Long division

(long)Division tableauSchoolbook long division
In order to convert a rational number represented as a fraction into decimal form, one may use long division.
Otherwise, it is still a rational number but not a b-adic rational, and is instead represented as an infinite repeating decimal expansion in base b positional notation.

Full reptend prime

full repetend prime
A prime is a proper prime if and only if it is a full reptend prime and congruent to 1 mod 10.
Base 10 may be assumed if no base is specified, in which case the expansion of the number is called a repeating decimal.

Series (mathematics)

infinite seriesseriespartial sum
A repeating decimal can also be expressed as an infinite series.
For instance, a recurring decimal, as in

Duodecimal

base 1212base-12
For example, in duodecimal, 1/2 = 0.6, 1/3 = 0.4, 1/4 = 0.3 and 1/6 = 0.2 all terminate; 1/5 = 0.2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1/7 = 0.186ᘔ35 has period 6 in duodecimal, just as it does in decimal.
Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a terminating representation in duodecimal.

Unique prime

unique period
A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q.

Positional notation

positionalpositional numeral systemplace-value
For more general fractions and bases see the algorithm for positive bases.

Fermat's little theorem

Fermat's TheoremFermat little theoremFermat's "little theorem
This result can be deduced from Fermat's little theorem, which states that 10 p−1 ≡ 1 (mod p).

Parasitic number

Dyson numberparasitic numbers
Notice that the repeating decimal

Midy's theorem

In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period.

Balanced ternary

balancedternary
The conversion of a repeating balanced ternary number to a fraction is analogous to converting a repeating decimal.

Decimal representation

decimal expansiondecimaldecimal expression
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.

Numerical digit

digitdigitsdecimal digit
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.

Periodic function

periodicperiodperiodicity
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.

Infinity

infiniteinfinitely
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.

0

zerozero function0 (number)
A repeating or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero.

Decimal separator

decimal pointdecimal markthousands separator
For example, the decimal representation of 1⁄3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227⁄555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals.

Divisor

divisibilitydividesdivisible
Every terminating decimal representation can be written as a decimal fraction, a fraction whose divisor is a power of 10 (e.g. ); it may also be written as a ratio of the form k⁄2 n 5 m (e.g. ). However, every number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit 9.

Exponentiation

exponentpowerpowers
Every terminating decimal representation can be written as a decimal fraction, a fraction whose divisor is a power of 10 (e.g. ); it may also be written as a ratio of the form k⁄2 n 5 m (e.g. ). However, every number with a terminating decimal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit 9.